Volume 35, Issue 2. Subsidy or tax policy for new technology adoption in duopoly with quadratic and linear cost functions

Transcription

1 Volume 35, Issue 2 Subsidy or tax policy for new technology adoption in duopoly with quadratic and linear cost functions Masahiko Hattori Faculty of Economics, oshisha University Yasuhito Tanaka Faculty of Economics, oshisha University Abstract We present an analysis about subsidy (or tax) policy for adoption of new technology in a duopoly with a homogeneous good. Technology itself is free. However, firms must expend fixed set-up costs for adoption of new technology, for example, education costs of their staffs. We assume linear demand function, and consider two types of cost functions of firms. Quadratic cost functions and linear cost functions. There are various cases of optimal policies depending on the level of the set-up cost and the forms of cost functions. In particular, under linear cost functions there is the following case. The social welfare is maximized when one firm adopts new technology, however, both firms adopt new technology without subsidy nor tax. Then, the government should impose taxes on one firm or both firms. Under quadratic cost functions there exists no taxation case. There are subsidization cases both under quadratic and linear cost functions. We thank the referee for giving valuable comments which substantially improved this paper. Citation: Masahiko Hattori and Yasuhito Tanaka, (2015) ''Subsidy or tax policy for new technology adoption in duopoly with quadratic and linear cost functions '', Economics Bulletin, Volume 35, Issue 2, pages Contact: Masahiko Hattori - eeo1101@mail3.doshisha.ac.jp, Yasuhito Tanaka - yasuhito@mail.doshisha.ac.jp. Submitted: February 17, Published: June 0, 2015.

2 1 Introduction We present an analysis about subsidy (or tax) policy for adoption of new technology in a duopoly with a homogeneous good. Technology itself is free. However, irms must expend ixed set-up costs for adoption of new technology, for example, education costs of their staffs. We assume linear demand function, and consider two types of cost functions of irms. Quadratic cost functions and linear cost functions. With quadratic cost functions marginal costs are increasing, and with linear cost functions marginal costs are constant. There are several references about technology adoption or R& in duopoly or oligopoly. Lots of researches focus on the relation between technology licensor and licensee. The difference of means of contracts which are royalties, up-front fees, the combinations of these two and auction are well discussed(katz and Shapiro (185), Kamien and Tauman (186), Sen and Tauman (2007)). Kamien and Tauman (186) shows that if the licensordoesnothaveproductioncapacity,ixedfee isbetterthan royaltyanditisalso better for consumers. This topic under Stackelberg oligopoly is discussed in Kabiraj (2004) when the licensor does not have production capacity, and discussed in Wang and Yang (2004), Kabiraj (2005) and Filippini (2005) when the licensor has production capacity. A Cournot oligopoly with ixed fee under cost asymmetry is analyzed in La Manna (13). He shows that if technologies can be replicated perfectly, a lower-cost irm has always the incentive to transfer its technology and hence a Cournot-Nash equilibrium cannot be fully asymmetric, but there exists no non-cooperative Nash equilibrium in pure strategies. On the other hand, using cooperative game theory, Watanabe and Muto (2008) analyses bargaining between licensor with no production capacity and oligopolistic irms. In recent research, the relation between market structure and technology improvement is analyzed. Boone (2001) and Matsumura et. al. (2013), respectively, ind a non-monotonic relation between intensity of competition and innovation. Also, Pal (2010) shows that if we consider technology adoption, Cournot competition makes more social welfare than Bertrand competition under differentiated goods market. Elberfeld and Nti (2004) examines the adoption of a new technology in oligopoly, where there is ex-ante uncertainty aboutvariablecostsofthenewtechnology,andshowsthatifinequilibriumbotholdand new technologies are employed, more uncertainty about the new technology increases (decreases) the number of innovating irms and decreases (increases) the product price if the up-front investment is large (small). Zhang et. al. (2014) analyzes the effect of information spillovers when the outcome of R& is uncertain in a two-stage Cournot oligopoly model where a subset of irms irst make a choice between two alternative production technologies independently and then all irms compete in quantity. This paper analyzes optimal subsidization or taxation policies about adoption of new technology by irms in a duopoly with a homogeneous good. We consider the following three-stage game. (1) The irst stage: The government determines the level of subsidies (or taxes) to the irms. (2) The second stage: The irms decide whether they adopt new technology or not.

3 (3) The third stage: The irms determine their outputs. Thesocialwelfareisdeinedtobeconsumers surplusplusirms proits,whichisequalto consumers utility minus productions costs including the set-up costs of new technology. Subsidies to the irms are inanced by lump-sum taxes on the consumers, and revenues fromtaxesontheirmsaretransferedtotheconsumersinalum-summanner. Theselumpsum taxes and transfers are not related to the good of this industry. Excluding income effectstheydonotaffectthedemandforthegood,andtheyarecanceledoutinthesocial welfare. There are various cases about optimal policies depending on the level of the set-up cost and the forms of cost functions. Under quadratic cost functions there are the following cases. (1) The social welfare is maximized when both irms adopt new technology, but only one irm adopts new technology without subsidy. Then, the government should give subsidies to the irms. There are two subsidization schemes. a) The government gives a subsidy to one of the irms, and this irm adopts new technology. The other irm does not adopt new technology. It is a discriminatory policy. b) The government gives chances to receive subsidies to both irms, but actually onlyoneirmreceivesasubsidyandadoptsnewtechnology. Itisnotadiscriminatory policy. In both schemes only one irm adopts new technology. (2) The social welfare is maximized when both irms adopt new technology, and they adopt new technology without subsidy. Then, the government should do nothing. Under linear cost functions there are the following cases. (1) The social welfare is maximized when one irm adopts new technology, however, both irms adopt new technology without subsidy nor tax. Then, the government should impose taxes for new technology adoption on one irm, or both irms. There are the following two taxation schemes. a) The government imposes a tax on one irm to prevent adoption of new technology. It is a discriminatory policy. b) The government imposes taxes on both irms. At the equilibrium only one of the irms adopts new technology and this irm actually pays a tax. The other irm does not adopt. It is not a discriminatory policy because adoption of new technologyisachoiceof theirm. In both schemes only one irm adopts new technology. (2) The social welfare is maximized when one irm adopts new technology, and only one irm adopts new technology without subsidy nor tax. Then, the government should do nothing.

4 Under quadratic cost functions there exists no taxation case. Note that our model is (at least mathematically) equivalent to a model of technology license with a ixed license fee. 2 Themodel Two irms, Firm A and B, produce a homogeneous good, and consider adoption of new technology from a foreign country. Technology itself is free, but each irm must expend a ixed set-up cost, for example, education cost of its staff. enote the outputs of Firm A andbby x A and x B,theprice of thegood by p. Theutilityfunctionof consumersis u a.x A C x B / 1 2.x A C x B / 2 ; where a is a positive constant. The inverse demand function is derived as follows. p a x A x B : (1) Under quadratic cost functions the cost functions of the irms before adoption of new technology are cxi 2 ; i A; B, and the cost functions after adoption of new technologyare 1 2 cx2 i ; i A:B. A ixed set-upcostis e. (2) Under linear cost functions the cost functions of the irms before adoption of new technologyare cx i ; i A; B, andthecostof each irm after adoption of new technology iszero. A ixed set-upcostisalso e. c in both cost functions and e are positive constants and common to both irms. There existsno ixed costotherthan theset-upcosts. The social welfare W is deined to be the sum of consumers surplus and irms proits, which is equal to consumers utility minus productions costs including the set-up costs of new technology, as follows; W a.x A C x B / a.x A C x B / 1 2.x A C x B / 2 p.x A C x B / C Œp.x A C x B / c A.x A / c B.x B / 1 2.x A C x B / 2 c A.x A / c B.x B / c A.x A /and c B.x B /generallydenotethecostfunctionsofirms. Theymayincludeset-up costs. Subsidies to the irms are inanced by lump-sum taxes on the consumers, and revenues fromtaxesontheirmsaretransferedtotheconsumersinalum-summanner. Theselumpsum taxes and transfers are not related to the good of this industry. Excluding income effectstheydonotaffectthedemandforthegood,andtheyarecanceledoutinthesocial welfare. We analyze the optimal subsidization or taxation policies of the government for adoption of new technology by irms. If adoption of new technology and non-adoption are indifferent for a irm, then it adopts new technology.

5 3 Quadratic cost functions In this section irms have quadratic cost functions. The proits of Firm A and B before adoption of new technology are A.a x A x B /x A cx 2 A ; B.a x A x B /x B cx 2 B : After adoption of new technology they are A.a x A x B /x A 1 2 cx2 A e; B.a x A x B /x B 1 2 cx2 B e: We assume Cournot type behavior of irms. There are four cases. (1) The conditions for proit maximization when no irm adopts new technology are a 2x A x B 2cx A 0; a x A 2x B 2cx B 0: We denote the equilibrium outputs as follows; and the equilibrium proits by x 0 A x0 B a 2c C 3 ; 0 A 0 B.c C 1/a2.2c C 3/ 2 : (2) The conditions for proit maximization when both irms adopt new technology are a 2x A x B cx A 0; a x A 2x B cx B 0: We denote the equilibrium outputs as follows; and the equilibrium proits by Qx A Qx B Q A Q B a c C 3 :.c C 2/a2 2.c C 3/ 2 e: (3) If only Firm A adopts new technology, the conditions for proit maximization are a 2x A x B cx A 0; a x A 2x B 2cx B 0: We denote the equilibrium outputs as follows; x A A and the equilibrium proits by.c C 2/a2 2c 2 C 6c C 3 ;.c C 1/a xa B 2c 2 C 6c C 3 : A A.c C 2/.2c C 1/2 a 2 2.2c 2 C 6c C 3/ 2 e; A B.c C 1/3 a 2.2c 2 C 6c C 3/ 2 :

6 (4) If only Firm B adopts new technology, the equilibrium outputs are written as; x B A and the equilibrium proits as.c C 1/a 2c 2 C 6c C 3 ; xb B.c C 2/a2 2c 2 C 6c C 3 : B A.c C 1/3 a 2.2c 2 C 6c C 3/ 2 ; B B.c C 2/.2c C 1/2 a 2 2.2c 2 C 6c C 3/ 2 e: Let e 1 Q A C e B A Q B C e A B.2c4 C 14c 3 C 36c 2 C 40c C 15/a 2 c 2.2c C 3/ 2.2c 2 C 6c C 3/ 2 ; e 0 A A C e 0 A B B C e 0 B.8c4 C 40c 3 C 72c 2 C 56c C 15/a 2 c 2.2c C 3/ 2.2c 2 C 6c C 3/ 2 : If e e 1, the best response to adoption is adoption. If e > e 1, the best response to adoption is non-adoption. If e e 0, the best response to non-adoption is adoption. If e > e 0, thebestresponseto non-adoption isnon-adoption. Weind e 0 e 1.c C 2/.8c3 C 38c 2 C 54c C 27/c 2 a 2 2.c C 3/ 2.2c C 3/ 2.2c 2 C 6c C 3/ 2 > 0: The game after the second stage is depicted as follows. B adoption of new technology non-adoption adoptionof Q B e B A A newtechnology Q A e A A e B B e B 0 non-adoption B A The sub-game perfect equilibria are as follows. Lemma 1. (1) If e e 1,thesub-gameperfectequilibriumisastatesuchthatbothirms adoptnewtechnology.inthiscase e e 1 and e e 0,soadoptionofnewtechnology isadominantstrategyforeachirm. (2) If e 1 < e e 0,thesub-gameperfectequilibriumisastatesuchthatoneirm,Firm AorB,adoptsnewtechnology. Inthiscase e e 0 and e > e 1,soadoptionofnew technology is a best response to non-adoption, and non-adoption is a best response to adoption. (3) If e > e 0, the sub-game perfect equilibrium is a state such that no irm adopts new technology. Inthiscase e > e 0 and e > e 1,sonon-adoptionisadominantstrategy for both irms. 0 A

7 Socialwelfare enotethesocialwelfarewhenbothirmsadoptnewtechnologyby W 2, that when one irm adopts new technology by W 1 and that when no irm adopts new technologyby W 0. Then,we have W 2.c C 4/a2.c C 3/ 2 2e; W 1.6c3 C 27c 2 C 27c C 8/a 2 2.2c 2 C 6c C 3/ 2 e; W 0 2.c C 2/a2.2c C 3/ 2 : Let e 0 a.8c4 C 52c 3 C 102c 2 C 71c C 15/a 2 c 2.2c C 3/ 2.2c 2 C 6c C 3/ 2 ; e 1 a.2c4 C 17c 3 C 45c 2 C 43c C 15/ 2.c C 3/ 2.2c 2 C 6c C 3/ 2 : We have W 0 W 1 when e ea 0, W 1 W 2 when e ea 1, W 0 > W 1 (or W 1 > W 0 ) when e > ea 0 (or e < e0 a ) and W 1 > W 2 (or W 2 > W 1 ) when e > ea 1 (or e < e1 a ). We can show ea 0 ea 1.c2 C 7c C /.4c 2 C 14c C /a 2 c 2.c C 3/ 2.2c C 3/ 2.2c 2 C 6c C 3/ 2 > 0: Thus, the following lemma is derived. Lemma 2. (1) If e e 1 a, W 2 isthemaximum,andadoptionofnewtechnologybyboth irms is optimal. (2) If e 1 a < e e0 a, W 1 isthemaximum,andadoptionofnewtechnologybyoneirmis optimal. (3) If e > e 0 a, W 0 isthemaximum,andnon-adoptionofnewtechnologybyanyirmis optimal. Now we ind e 1 a e 0.4c4 C 18c 3 C 17c 2 18c 27/a 2 c 2 2.c C 3/ 2.2c C 3/ 2.2c 2 C 6c C 3/ 2 : This is positive for reasonable value of c. For example, if c > 1:07, ea 1 obtain the following results. e 0 > 0. Then, we Theorem 1. Under quadratic cost functions the optimal policies should be as follows. (1) If e e 1, W 2 isoptimalandbothirmsadoptnewtechnologywithoutsubsidy. The government should do nothing. (2) If e 1 < e e 0, W 2 isoptimalbutoneirmadoptsnewtechnologywithoutsubsidy. Thegovernmentshouldgivesubsidiestobothirms.Thelevelofthesubsidymustnot besmallerthan e e 1. Ifthegovernmentgivesasubsidytoonlyoneirm,theother irm does not adopt new technology. Thus, in this case the government should give the subsidies to both irms.

8 (3) If e 0 < e e 1 a, W 2 isoptimalandnoirmadoptsnewtechnologywithoutsubsidy. Thegovernmentshouldgivesubsidiestobothirms.Thelevelofthesubsidymustnot besmallerthan e e 1. (4) If e 1 a < e e0 a, W 1 isoptimalbutnoirmadoptsnewtechnologywithoutsubsidy.the government should give subsidies to the irms. There are two subsidization schemes. a) The government gives a subsidy to one of the irms, and this irm adopts new technology. Thelevelofthesubsidymustnotbesmallerthan e e 0. Theother irm does not adopt. It is a discriminatory policy. b) Thegovernmentgivessubsidiestobothirms. Thelevelofthesubsidytoeach irmisbetween e e 0 and e e 1.Sinceattheequilibriumonlyoneirmadopts new technology, the government actually gives the subsidy to one of the irms. It is not a discriminatory policy because both irms have chances to receive subsidies. In both schemes only one irm adopts new technology. (5) If e > e 0 a, W 0 is optimal and no irm adopts new technology without subsidy. The government should do nothing. 4 Linear cost functions In this section we assume that irms have linear cost functions. Then, the marginal costs are constant. We use the same symbols as those in the previous section. The proits of FirmAandBbeforeadoption of new technologyare A.a x A x B /x A cx A ; B.a x A x B /x B cx B : After adoption of new technology they are A.a x A x B /x A e; B.a x A x B /x B e: Weassume Cournottype behaviorof irms,andassume a > 2c. Therearefour cases. (1) The conditions for proit maximization when no irm adopts new technology are a 2x A x B c 0; a x A 2x B c 0: The equilibrium outputs and proits are x 0 A x0 B a c 3 ; 0 A 0 B.a c/2 : (2) The conditions for proit maximization when both irms adopt new technology are The equilibrium outputs and proits are a 2x A x B 0; a x A 2x B 0: Qx A Qx B a 3 ; Q A Q B a2 e:

9 (3) If only Firm A adopts new technology, the conditions for proit maximization are The equilibrium outputs and proits are a 2x A x B 0; a x A 2x B c 0: x A A a C c 3 ; xb A a 2c ; A.a C c/2 A 3 e; A B.a 2c/2 : (4) If only Firm B adopts new technology, the equilibrium outputs and proits are x B A a 2c 3 ; xb B a C c ; A B 3.a 2c/2 ; B B.a C c/2 e: Let e 1 Q A C e A B Q B C e B A 4.a c/c ; e 0 A A C e 0 A B B C e 0 B 4ac : Clearly, e 0 > e 1. Thus,similarlytoLemma1thesub-gameperfectequilibriaofthegame after the second stage are as follows. Lemma 3. (1) If e e 1,thesub-gameperfectequilibriumisastatesuchthatbothirms adoptnewtechnology.inthiscase e e 1 and e e 0,soadoptionofnewtechnology isadominantstrategyforeachirm. (2) If e 1 < e e 0,thesub-gameperfectequilibriumisastatesuchthatoneirm,Firm AorB,adoptsnewtechnology. Inthiscase e e 0 and e > e 1,soadoptionofnew technology is a best response to non-adoption, and non-adoption is a best response to adoption. (3) If e > e 0, the sub-game perfect equilibrium is a state such that no irm adopts new technology. Inthiscase e > e 0 and e > e 1,sonon-adoptionisadominantstrategy for both irms. Social welfare The social welfare are obtained as follows; Let W 2 4a2 2e; W 1 8a2 8ac C 11c 2 18 e 0 a.8a C 3c/c ; ea 1 18 e; W 0.8a 11c/c : 18 4.a c/2 : Then, W 0 W 1 when e ea 0, W 1 W 2 when e ea 1, W 0 > W 1 (or W 1 > W 0 ) when e > ea 0 (or e < e0 a ) and W 1 > W 2 (or W 2 > W 1 ) when e > ea 1 (or e < e1 a ). Clearly, ea 0 > e1 a. Similarly to Lemma 2we get thefollowing lemma.

10 Lemma 4. (1) If e e 1 a, W 2 isthemaximum,andadoptionofnewtechnologybyboth irms is optimal. (2) If e 1 a < e e0 a, W 1 isthemaximum,andadoptionofnewtechnologybyoneirmis optimal. (3) If e > e 0 a, W 0 isthemaximum,andnon-adoptionofnewtechnologybyanyirmis optimal. Comparing e 0, e 1, ea 0 and e1 a,we ind e 1 a < e1 < e 0 < e 0 a : This is different from the result in the previous section. Under quadratic cost functions we have e 1 < e 0 < e 1 a < e0 a. We obtain the following results. Theorem 2. Under linear cost functions the optimal policies should be as follows. (1) If e e 1 a, W 2 isoptimalandbothirmsadoptnewtechnologywithoutsubsidynor tax. The government should do nothing. (2) If e 1 a < e e1, W 1 isoptimalbutbothirmsadoptnewtechnologywithoutsubsidy nor tax. The government should impose taxes for new technology adoption to one irm, or to both irms. There are the following two taxation schemes. a) The government imposes a tax on one irm, Firm A or B, to prevent adoption of new technology. The level of the tax must be larger than e 1 e. It is a discriminatorypolicy.whenoneoftheirmsdoesnotadopt,theotherirmhasan incentive to adopt. b) Thegovernmentimposestaxesonbothirms. Thelevelofthetaxoneachirm isbetween e 1 e and e 0 e. Then,attheequilibriumoneoftheirmsadopts newtechnologyandthisirmactuallypaysatax.theotherirmdoesnotadopt. It is not a discriminatory policy because adoption of new technology is a choice oftheirm. In both schemes only one irm adopts new technology. (3) If e 1 < e e 0, W 1 is optimal and only one irm adopts new technology without subsidy nor tax. The government should do nothing. (4) If e 0 < e ea 0, W 1 is optimal but no irm adopts new technology without subsidy nortax.thegovernmentshouldgivesubsidiestobothirms.thelevelofthesubsidy must not be smaller than e e 0. Since only one irm adopts new technology with thislevelofthesubsidy,thegovernmentactuallygivesthesubsidytooneoftheirms which adopts new technology. However, it gives chances to receive subsidies to both irms. It is not a discriminatory policy. Similarly to Theorem 1 there exists another discriminatory subsidization scheme such thatthegovernmentgivesasubsidytoonlyoneirm.

11 (5) If e > e 0 a, W 0 isoptimalandnoirmadoptsnewtechnologywithoutsubsidynortax. The government should do nothing. Remarkable results are (2) and (3) of this theorem. In (2) adoption of new technology by only one irm is optimal for the society, however both irms have incentives to adopt new technology. Thus, the government must impose taxes on one of the irms or to both irms so as to prevent adoption by one irm. On the other hand, in (2) of Theorem 1 adoptionofnewtechnologybybothirmsisoptimal,butonlyoneirmhasanincentiveto adopt. Thus,thegovernmentgivesubsidiestotheirms. In(3)ofthistheoremadoptionof newtechnologybyoneirmisoptimal,andonlyoneirmhasanincentivetoadopt. Thus, the government should do nothing. On the other hand, in (3) of Theorem 1 adoption of new technology by both irms is optimal, however no irm has an incentive to adopt. Therefore, the government give subsidies to the irms. (1), (4) and (5) are the same as those in Theorem 1. In the future research we want to generalize the analyses in this paper to a case of general demand and cost functions. Acknowledgment The authors would like to thank the referee for his/her valuable comments which helped to improve the manuscript. References Boone, J. (2001), Intensity of competition and the incentive to innovate, International Journal of Industrial Organization, 1, pp Elberfeld, W. and Nti, K. O.(2004), Oligopolistic competition and new technology adoption under uncertainty, Journal of Economics, 82, pp Filippini, L. (2005), Licensing contract in a Stackelberg model, The Manchester School, 73, pp Hattori, M. and Y. Tanaka(2014), Incentive for adoption of new technology in duopoly under absolute and relative proit maximization, Economics Bulletin, 34. pp Kabiraj, T.(2004), Patent licensing in a leadership structure, The Manchester School, 72, pp Kabiraj, T. (2005), Technology transfer in a Stackelberg structure: Licensing contracts and welfare, The Manchester School, 73, pp Kamien, T. and Tauman, Y.(186), Fees versus royalties and the private value of a patent, Quarterly Journal of Economics, 101, pp

12 Katz, M. and Shapiro, C. (185), On the licensing of innovations, Rand Journal of Economics, 16, pp La Manna M.(13), Asymmetric Oligopoly and Technology Transfers, Economic Journal, 103, pp Matsumura, T., Matsushima, N. and Cato, S.(2013), Competitiveness and R& competition revisited, Economic modelling, 31, pp Pal, R. (2010), Technology adoption in a differentiated duopoly: Cournot versus Bertrand, Research in Economics, 64, pp Sen,. and Tauman, Y. (2007), General licensing schemes for a cost-reducing Innovation, Games and Economic Behavior, 5, pp Wang, X. H. and Yang, B. Z. (2004), On technology licensing in a Stackelberg duopoly, Australian Economic Papers, 43, pp Watanabe N. and Muto, S. (2008), Stable proit sharing in a patent licensing game: general bargaining outcomes, International Journal of Game Theory, 37, pp Zhang, Y., Mei, S. and Zhong, W. (2014), New technology adoption in a Cournot oligopoly with spillovers, Journal of Economics, 112, pp